We construct a simple model for stationary, axisymmetric black-hole magnetospheres, in which the poloidal magnetic field is generated by a toroidal electric current in a thin disk with the inner edge, by solving the vacuum Maxwell equations in Schwarzschild background. In this work, to obtain a concise analytical form of the magnetic stream function, we use the approximation that the inner edge is far distant from the event horizon. The global magnetospheric structure with the closed-loop and open field lines threading the inner and outer parts of the disk is explicitly shown, claiming that the model is useful for studying astrophysical problems in relation to accretion flows onto a black hole and disk winds to infinity. The asymptotic shape of the field lines at the event horizon becomes nearly cylindrical, while at infinity it becomes conical. The magnetic spot in the disk connected with the black hole through the loop field lines occupies a very narrow region with the ring area roughly equal to the horizon area. By taking account of the existence of a uniform (external) magnetic field, we also obtain the model allowing the collimation of field lines. Then, it is found that the magnetic connection between the black hole and the disk breaks down if the uniform field is strong enough. The final discussion is devoted to the generation of a toroidal magnetic field due to slow rotation of the magnetosphere and the screw instability as a trigger of flare-like energy release.
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